What happens after the results of the election, since at least one Senate seat will be open. Does a special election occur or does someone appoint a replacement?

In most instances, the Governor of the state appoints someone and frequently that person serves until the next regular election. But it is gfoverned by the 17th amendment partially desribed in this excerpt from a writeup about the appointments:

Procedures Governing Vacancies
Vacancies in Congress occur when a Senator or Representative dies, resigns, declines
To serve, or is expelled or excluded from either house.
Senate
Procedures governing vacancies in the Senate were initially established by Article
I, Section 3 of the Constitution, as later amended by paragraph 2 of the 17th Amendment.
The latter states:
CRS-2
1 Or. Rev. Stat. §188.120 (2001).
2 Wis. Stat. § 17.18 (1999-2000).
3 Okla. Stat. tit. 26, §12-101.
4 Alaska Stat. §15.40.010 (2001); Ariz. Rev. Stat. §16.222 (2001); and Haw. Rev. Stat. § 17-1
(2001).
5 Utah Code Ann. § 20A-1-502(2) (2001) and Wyo. Stat. § 22-18-111 (I) (2002).
When vacancies happen in the representation of any State in the Senate, the executive
Authority of such State shall issue writs of election to fill such vacancies: Provided
That the legislature of any State may empower the executive thereof to make
Temporary appointments until the people fill the vacancies by election as the
Legislature may direct.
Appointment of Interim Senators. Prevailing practice is for state governors to
Fill Senate vacancies by appointment, with the appointee serving until a special election
Has been held, at which time the appointment expires immediately. In the event a seat
Becomes vacant between the time of a general election and the expiration of the term,
However, the appointee usually serves the balance of the term, until the next regularly
Scheduled general election. This practice originated with the constitutional provision that
Applied prior to the popular election of senators, under which governors were directed to
Make temporary appointments when state legislatures were in recess. It was intended to
Ensure continuity in a state's Senate representation during the lengthy intervals between
State legislative sessions.
The governor's direct authority to make interim appointments is specified in the
Various state laws. Oregon1 and Wisconsin2 do not allow the governor to make interim
Appointments, requiring, instead, a special election to fill any Senate vacancy. The State
Of Oklahoma also requires that Senate vacancies be filled by special elections, with an
Exception. If the vacancy occurs after March 1 of any even-numbered year and the term
Expires the following year, no special election is held; rather, the governor is required to
Appoint the candidate elected in the regular general election to fill the unexpired term.3
At least five states restrict the governor's power to appoint interim Senators. Alaska,
Arizona, and Hawaii require the governor to fill Senate vacancies with a person affiliated
With the same political party as the previous incumbent.4 Utah and Wyoming require the
Governor to select an interim senator from a list of three candidates proposed by the state
Central committee of the political party with which the previous incumbent was affiliated.5
Many states limit the term of office for interim senators to the date set for the special
Election. In these cases, the term of the interim senator expires immediately upon the
Election of the popularly chosen successor, who serves the balance of the Senate term,
Whether it is a few weeks or several years. Moreover, when an interim appointment is
Made late in the term, it is often customary for the interim senator to resign his or her seat
Immediately after the election, and for the governor to appoint the special election winner
To serve the balance of the term. It is also customary, for the purposes of determining
Seniority, for the newly elected replacement senator to be sworn in as soon as possible.

Agree with smearcase

Just wanted to add that Sarah Palin will soon be appointing a replacement for the re-elected but convicted .Senator Stevens of Alaska.

My guess is that she appoints herself to the office and resigns as Governor .

You heard it here 1st!

Illinois Governor Rod Blagojevich gets to pick someone for Obama's unexpired term - Obama was only 4 years into his first term as Senator. Don't be overly surprised if Blagojevich picks himself for the seat! And Delaware Governor Minner gets to appoint someone for Joe Biden's spot - on election day Biden was re-elected to a new 6 year term, which of course he won't be serving.

Blagojevich may be serving from jail if Fed Prosecutor Patrick Fitgerald does his job properly .

These are not the only ones. NJ has a habit of playing this game. They just re-elected Frank Lautenberg who is older than Methuselah .He will retire and former Senator Jon Corzine now Governor may return to the Senate if Obama doesn't pick him for Treasury .
JF Kerry could be picked for Sec State and then Obama buddy Governor Deval Patrick of Mass. Would get to pick his replacement . And on and on and on it goes .